#153323

order	512
nilpotency class	5
isoclinism	isoclinic class of #153078
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6,9,11)(7,12,8,10)(13,14)(15,20)(16,21)(17,23)(18,32)(19,33)(22,27)(24,28)(25,36)(26,40)(29,37)(30,34)(31,44)(35,42)(38,41)(39,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,17,19)(14,20,23,33)(16,25,29,30)(18,26,31,28)(21,36,37,34)(22,35,39,38)(24,32,40,44)(27,42,43,41);(2,4)(5,8,9,7)(6,10,11,12)(14,21)(15,19)(16,27)(18,28)(20,34)(22,24)(23,37)(25,41)(26,31)(29,43)(30,42)(32,38)(33,36)(35,44)(39,40)

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=d,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cf,e^c=eg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60519);